Contractors’ Go/No-go Decision

This section explains two different methods to present contractors’ different risk-taking behaviors in their go/no-go decision. In the conventional method using expected utility, different risk attitudes are classified into three generic types: averse, risk-neutral, and risk-seeking. Meanwhile, the proposed new method describes different risk attitudes of contractors by different amounts of maximum loss allowances, which represent various degrees of risk attitude, i.e., how much risk-averse or risk-tolerant a contractor is.

5.5.1 Go/No-go Decision Using Utility Functions

Consider a contractor that decides the fixed price of a job before performing the job. The actual cost of the job is subject to uncertainties at the time of decision and the true cost will be determined at the completion of the job. The contractor is responsible for all cost risks (actual cost greater than the price) and the owner takes no cost risk. The contractor’s price setting is a risky task since the contractor does not know the actual cost of the job. But, the contractor has some knowledge about possible range of actual cost and probability, i.e., a probability distribution of actual cost, as in Figure 5.4.

Let the expected cost of the job be E(x), which is based on contractors’

probability distributions. In the go/no-go decision for a job, a risk-neutral contractor would proceed to bid at a bid equal to E(x), which has a 50% chance of gain and a 50%

chance of loss. The contractor’s go/no-go decision simply depends on E(x), which is

based on expected monetary value theory. The contractor would not care about downside and upside uncertainties.

However, risk-averse or risk-seeking contractors would have different

perceptions on the downside and upside uncertainties. A risk-averse contractor would proceed to bid if its bid is greater than E(x) + its risk premium. On the other hand, a risk-seeking contractor would proceed to bid unless its bid amount is smaller than E(x) – its risk premium. The amount of risk premium varies depending on the degrees of risk-aversion or risk-seeking of the contractors. So, contractors having different risk attitudes would have different budget thresholds in their bid decision, which can be represented by E(x) ± risk premium.

Utility functions can be used to represent these contractors’ different risk attitudes and to quantify the magnitude of risk premiums. In the following description, for simplicity of comparison, it is assumed that all jobs are identical and bid decisions by contractors are independent and repetitive. Then, contractors’ utility functions need to be normalized so that possible values of utility lie between utility zero and utility 1.0.

This normalization is a common method to apply utility functions for different individuals evaluating the same subject. Equation 5.10 shows the normalized

exponential utility function. The shape of utility functions varies depending on the value of risk-aversion coefficient r as listed in Table 5.1.

1 exp( ( ) )

Where, r is the risk-aversion coefficient to define risk attitude;

x is the wealth (contractor’s bid) and x > 0;

a is the minimum value of wealth (associated with the minimum bid); and b is the maximum value of wealth (associated with the maximum bid).

Table 5.1 Risk-aversion Coefficient and Characteristic of Utility Functions

Value of coefficient Curvatures Risk attitudes

r > 0 Concave Risk-averse

r = 0 Straight Risk-neutral

r < 0 Convex Risk-seeking

Figure 5.5 provides an example of the normalized utility functions. The assumed value of expected cost E[x] is $10M. A contractor’s minimum bid that attains the minimum utility (0.0) and the maximum bid that attains the maximum utility (1.0) are assumed to be $6.0M (the value of a) and $14.0M (the value of b) in Equation 5.10, respectively. As shown in Figure 5.5, positive values of the risk-aversion coefficient develop concave curves, indicating risk-aversion. In contrast, negative values of the coefficient develop convex curves, indicating risk-seeking.

Individual contractors could have different estimates for an identical job even if they have similar levels of estimating capability. A contractor’s bid may take any value within the ranges specified on the x axis in Figure 5.5.

0.0

Figure 5.5 Example of Normalized Exponential Functions

Calculation of Risk Premium

This section provides a detailed procedure for the estimation of the risk premium for contractors that have different risk attitudes using the normalized exponential utility functions as follows:

1) Define a contractor’s utility function by choosing a value of risk-aversion coefficient r.

2) Assume a probability distribution of actual cost and define the expected cost of the job, E(x) from the distribution. Also, define the maximum and the minimum values for wealth in Equation 5.10.

3) Calculate the utility of the expected cost, U[E(x)] using the normalized utility function with the specified coefficient r.

4) Calculate the expected value of the utility of x, E[U(x)] using the normalized utility function with the coefficient r = 0 (risk-neutral).

5) Calculate the certainty equivalent (CE) using the inverse function of the utility function. Equation 5.11 is the general form of the inverse function of the normalized exponential utility function defined in Equation 5.10.

[ ]

0, ( )

0, 1 {1 U( )(1 exp( ( ) )}

if r x y b a a

if r x a LN x b a r

r

= = − +

≠ = − − − − − [5.11]

6) Calculate the risk premium: RP = E(x) – CE

The value of the risk premium can be positive, negative, or zero, depending on the individual contractor’s risk attitude.

7) Calculate budget threshold: BT = E(x) + RP

Then, go/no-go decisions by an individual contractor can be made given the contractors’ bid amount considering the individual’s risk attitude. The decision is based on the comparison of the bid amount with the individual’s own budget threshold as below:

ƒ Proceed to bid, if the bid amount ≥ BT since the contractor perceives the risk acceptable.

ƒ Decline to bid, if the bid amount < BT since the contractor perceives the risk unacceptable.

5.5.2 Go/No-go Decision Using Value at Risk

Suppose a contractor has its budget for a job. The contractor’s bid amount will be its budget when it bids and wins the job. The contractor will suffer a loss if the actual cost of the job turns out to be greater than its budget (contract price). But, the contractor will gain a profit in the opposite case. So, there are uncertainties in the contractor’s decision; downside uncertainties of losses and upside uncertainties of gains.

VaR can be calculated as follow. First, consider a unit normal variate t . t is distributed as the normal variate with mean = 0 and standard deviation = 1. The unit normal probability density function is shown in Equation 5.12.

2/ 2

( ) 2

f e τ τ

π

= − [5.12]

For simplicity, connote this unit normal function as ( )φ t . Given some budget B, the probability α that actual cost overruns k is the area under the Unit Normal curve from t=Bto t= ∞ is

B ( )d

α =

∫

and the VaR for this unit normal is

2/ 2

We may take note of the identity:

u u

e du=e

∫

Letting u= −τ²/ 2, substituting this into Equation 5.14, and then applying Equation 5.15 gives:

Therefore, the Value at Risk function for the unit normal distribution is simply the value of the unit normal as shown in Equation 5.17. Figure 5.6 shows αand VaR for the unit normal distribution. In the figure, VaR increases as the budget amount decreases, and vice versa.

( )

VaR=φ B [5.17]

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.0 0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0 2.3 2.5 2.8 3.0 3.3 3.5 3.8 4.0 4.3 4.5 4.8 5.0

Budget B

Alpha, VaR Alpha

VaR

Figure 5.6 Alpha & VaR for Unit Normal Distribution

In most cases, normal variates such as cost are more interesting than unit normal variates. Suppose x to be the cost which follows a Normal distributionN( ,μ σ_{c c}²). Define the unit normal variate t as in Equation 5.18.

t x μ

σ

= − [5.18]

Then,x= +μ σt dx; =σdt

Suppose a budget B is k standard deviations above the mean. Then,

,

It was found that Value at Risk function for the unit normal distribution is simply the value of the unit normal as shown in Equation 5.17. Substituting k with the budget B gives Equation 5.20:

Now, Value at Risk can be calculated for any normal distribution with mean and

Where, ( )Φ k is the cumulative unit normal probability function.

To make a go/no-go decision for a job, a contractor estimates Value at Risk based on its cost estimate. If VaR associated with the bid amount is greater than the contractor’s MLA, the contractor would not bid for the job because the contractor does not allow risks greater than its maximum loss allowance.

In the opposite case, if VaR associated with the bid amount is smaller than the contractor’s MLA, the contractor will bid because the contractor allows risks smaller than its maximum loss allowance. The contractor will bid while expecting potential profits under the assumption that the possible loss would not be significant from his own perspective: VaR < its own MLA. However, obviously, the contractor does not know the actual cost when it makes this decision.

Figure 5.7 simplifies the go/no-go decision discussed. Contractors decide to bid or not to bid depending on their own risk attitude that is represented by their own maximum loss allowances. The decision is based on the comparison between the

contractor’s own maximum loss allowance and the amount of expected loss from a job (Value at Risk), which is subject to their own estimates.

Estimating

Figure 5.7 Representation of Go/No-go Decision Making

Setting a target or reference points for decision makers in the proposed method for contractors’ go/no-go decision making is consistent with the ideas found in recent studies in behavioral decision theory. These studies maintain that individuals use their targets, or reference points in evaluating risky choices (Fiegenbaum and Thomas, 1988).

The proposed method is simple and easy for practitioners to understand compared to the alternate method using expected utility and concave or convex utility functions.